Solve 1/frac{1{x+10}-1/x}=720

$$\frac{ 1 }{ \frac{ 1 }{ x+10 } - \frac{ 1 }{ x } } = 720$$

Solve for x (complex solution)

$$x=-5+5\sqrt{287}i\approx -5+84.70537173i$$

$$x=-5\sqrt{287}i-5\approx -5-84.70537173i$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+10 and x is x\left(x+10\right). Multiply \frac{1}{x+10} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+10}{x+10}.

$$\frac{1}{\frac{x}{x\left(x+10\right)}-\frac{x+10}{x\left(x+10\right)}}=720$$

Since \frac{x}{x\left(x+10\right)} and \frac{x+10}{x\left(x+10\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{\frac{x-\left(x+10\right)}{x\left(x+10\right)}}=720$$

Do the multiplications in x-\left(x+10\right).

$$\frac{1}{\frac{x-x-10}{x\left(x+10\right)}}=720$$

Combine like terms in x-x-10.

$$\frac{1}{\frac{-10}{x\left(x+10\right)}}=720$$

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Divide 1 by \frac{-10}{x\left(x+10\right)} by multiplying 1 by the reciprocal of \frac{-10}{x\left(x+10\right)}.

$$\frac{x\left(x+10\right)}{-10}=720$$

Use the distributive property to multiply x by x+10.

$$\frac{x^{2}+10x}{-10}=720$$

Divide each term of x^{2}+10x by -10 to get -\frac{1}{10}x^{2}-x.

$$-\frac{1}{10}x^{2}-x=720$$

Subtract 720 from both sides.

$$-\frac{1}{10}x^{2}-x-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{10} for a, -1 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-\frac{1}{10}\right)\left(-720\right)}}{2\left(-\frac{1}{10}\right)}$$

Multiply -4 times -\frac{1}{10}.

$$x=\frac{-\left(-1\right)±\sqrt{1+\frac{2}{5}\left(-720\right)}}{2\left(-\frac{1}{10}\right)}$$

Multiply \frac{2}{5} times -720.

$$x=\frac{-\left(-1\right)±\sqrt{1-288}}{2\left(-\frac{1}{10}\right)}$$

Add 1 to -288.

$$x=\frac{-\left(-1\right)±\sqrt{-287}}{2\left(-\frac{1}{10}\right)}$$

Take the square root of -287.

$$x=\frac{-\left(-1\right)±\sqrt{287}i}{2\left(-\frac{1}{10}\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{287}i}{2\left(-\frac{1}{10}\right)}$$

Multiply 2 times -\frac{1}{10}.

$$x=\frac{1±\sqrt{287}i}{-\frac{1}{5}}$$

Now solve the equation x=\frac{1±\sqrt{287}i}{-\frac{1}{5}} when ± is plus. Add 1 to i\sqrt{287}.

$$x=\frac{1+\sqrt{287}i}{-\frac{1}{5}}$$

Divide 1+i\sqrt{287} by -\frac{1}{5} by multiplying 1+i\sqrt{287} by the reciprocal of -\frac{1}{5}.

$$x=-5\sqrt{287}i-5$$

Now solve the equation x=\frac{1±\sqrt{287}i}{-\frac{1}{5}} when ± is minus. Subtract i\sqrt{287} from 1.

$$x=\frac{-\sqrt{287}i+1}{-\frac{1}{5}}$$

Divide 1-i\sqrt{287} by -\frac{1}{5} by multiplying 1-i\sqrt{287} by the reciprocal of -\frac{1}{5}.

$$x=-5+5\sqrt{287}i$$

The equation is now solved.

$$x=-5\sqrt{287}i-5 x=-5+5\sqrt{287}i$$

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+10 and x is x\left(x+10\right). Multiply \frac{1}{x+10} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+10}{x+10}.

$$\frac{1}{\frac{x}{x\left(x+10\right)}-\frac{x+10}{x\left(x+10\right)}}=720$$

Since \frac{x}{x\left(x+10\right)} and \frac{x+10}{x\left(x+10\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{\frac{x-\left(x+10\right)}{x\left(x+10\right)}}=720$$

Do the multiplications in x-\left(x+10\right).

$$\frac{1}{\frac{x-x-10}{x\left(x+10\right)}}=720$$

Combine like terms in x-x-10.

$$\frac{1}{\frac{-10}{x\left(x+10\right)}}=720$$

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Divide 1 by \frac{-10}{x\left(x+10\right)} by multiplying 1 by the reciprocal of \frac{-10}{x\left(x+10\right)}.

$$\frac{x\left(x+10\right)}{-10}=720$$

Use the distributive property to multiply x by x+10.

$$\frac{x^{2}+10x}{-10}=720$$

Divide each term of x^{2}+10x by -10 to get -\frac{1}{10}x^{2}-x.

$$-\frac{1}{10}x^{2}-x=720$$

Multiply both sides by -10.

$$\frac{-\frac{1}{10}x^{2}-x}{-\frac{1}{10}}=\frac{720}{-\frac{1}{10}}$$

Dividing by -\frac{1}{10} undoes the multiplication by -\frac{1}{10}.

$$x^{2}+\left(-\frac{1}{-\frac{1}{10}}\right)x=\frac{720}{-\frac{1}{10}}$$

Divide -1 by -\frac{1}{10} by multiplying -1 by the reciprocal of -\frac{1}{10}.

$$x^{2}+10x=\frac{720}{-\frac{1}{10}}$$

Divide 720 by -\frac{1}{10} by multiplying 720 by the reciprocal of -\frac{1}{10}.

$$x^{2}+10x=-7200$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=-7200+5^{2}$$

Square 5.

$$x^{2}+10x+25=-7200+25$$

Add -7200 to 25.

$$x^{2}+10x+25=-7175$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=-7175$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{-7175}$$

Simplify.

$$x+5=5\sqrt{287}i x+5=-5\sqrt{287}i$$

Subtract 5 from both sides of the equation.

$$x=-5+5\sqrt{287}i x=-5\sqrt{287}i-5$$